198 Prof. Cayley on Differential Equations of the Second Order. 

 through the axis of revolution." Here 



The equation Aj9 = is integrable per se; and its integral is 

 » / C2 =c, or say y = c^l-t-p 2 , being, as is known, the dif- 

 ferential equation of a catenary having its directrix coincident 

 with the axis of revolution ; in fact the integral is 



c ■ y + c ' x+c ' 



y=2^ e c Jre c )• 



But there are difficulties as regards the application of this in- 

 tegral to the problem in hand; and Professor Challis is led to 

 consider that the solution of the problem " can be effected only 

 by taking into account an independent integration of A — 0;" 

 and this he proceeds to obtain by a method which " consists 

 essentially in first finding, when it is possible, the evolute of the 

 curve or curves of which A = is the differential equation, and 

 then employing the involutes thence derivable, which may be 

 regarded as the solution of the equation, to satisfy, either by 

 computation or by graphical construction, the given conditions 

 of the problem." This is perfectly allowable ; but after the evo- 

 lute is obtained, we must take not any involute, but the proper 

 involute of such evolute ; we thus have a solution of the differen- 

 tial equation A=0, the same as is obtained by what Professor 

 Challis considers to be the integration of the other equation 

 Aj9 = 0. This seems to me obvious a priori; but I will verify it 

 in regard to the problem in hand. Taking, with Professor 

 Challis, oc\ y ] as the coordinates of that point of the evolute which 

 is the centre of curvature at the point of the involute whose co- 

 ordinates are #, y, and writing also c~ 2k (the equation in ques- 

 tion is obtained in his paper), the several equations obtained by 

 him are in effect as follows : — 



y=W> 



~~~~ 2p' 



1 --.A" 2 _i 



(the third of these is his equation (a) , taking therein the lower 



